Maxwell's Twelve Equations

[BeholdTheTruth]

Can anyone tell me or point me to Maxwell's twelve equations?

Thanks, Yale L

[BillHoyt]

Originally posted by Yalel
Can anyone tell me or point me to Maxwell's twelve equations?

Thanks, Yale L

If you refer to Maxwell's equations relating electricity and magnetism, there are four such equations. You can find them in any good physics textbook. They are also here in cyberspace. Search for "maxwell equations".

Cheers,

[QuarkChild]

You could probably make twelve out of them if you tried...

1-4 Maxwell's differential equations in vacuum
5-8 Maxwell's integral equations in vacuum (same equations but a different form)
9-12 Maxwell's equations in media (same as 1-4 except for some extra terms)

Possibly 16 if the latter have an integral version as well.

[69dodge]

A vector equation in three dimensions corresponds to three scalar equations, one for each vector component.

I seem to recall that's how Maxwell himself wrote them.

[BeholdTheTruth]

Originally posted by BillHoyt


If you refer to Maxwell's equations relating electricity and magnetism, there are four such equations. You can find them in any good physics textbook. They are also here in cyberspace. Search for "maxwell equations".

Cheers,

Thanks, Bill. I've followed your suggestion and it seems that Maxwell had twenty equations that Heaviside boiled down into two. What I am looking for is the actual 20 equations that Maxwell started with, not what Heaviside ended up with, at according to http://occawlonline.pearsoned.com/bookbind/pubbooks/thomas_awl/chapter1/medialib/custom3/bios/heaviside.htm...

Heaviside, Oliver (1850--1925)

Heaviside studied electricity and languages on his own. He became a telegrapher and began working on results in electricity. He wrote two papers that caught the interest of Maxwell. Thereafter, Maxwell's work inspired Heaviside. Heaviside was able to simplify Maxwell's 20 equations into the two we now call Maxwell's equations. Heaviside's contributions in mathematics were in the areas of vector algebra and vector calculus. Using his methods he was able to efficiently solve systems of differential equations. He worked with Gibbs (in mathematics) and Thomson (in electricity).

[Edwin]

Two?

I always thought of it as four.

[BeholdTheTruth]

Originally posted by Edwin
Two?

I always thought of it as four.

Me as well, but per http://www.1729.com/topics/maxwellsEquations.html there seems to be some opinions that the four can be boiled down to two. OTOH, my question is what were the original twenty? Can anyone here answer that?

[Argo Nimbus]

Originally posted by Yalel
What I am looking for is the actual 20 equations that Maxwell started with, not what Heaviside ended up with

Maxwell originally worked with quaternions, so searching for maxwell + quaternions will produce some interesting links. In any case, Maxwell's original equations can be found in a pdf document here:

On the Notation of Maxwell's Field Equations, by Andre Waser (http://www.aw-verlag.ch/Documents/Notation%20of%20Maxwell%20Field%20Equations.PDF)

--- Argo

[peptoabysmal]

Why do you want Maxwell's original equations, Yalel? Looking perhaps for some "divine inspiration" to mathematically prove the existance of G.O.D.?

I see you got some new suckers to take the bait.

Let he who hath understanding, understand that this thread will lead nowhere except to some pointers to a very poorly made website with gibberish content.

[GreyWanderer]

Could anyone tell me the story behind these equations? How could he come up with these complex equations that could be simplified into two?

[Argo Nimbus]

Originally posted by 69dodge
A vector equation in three dimensions corresponds to three scalar equations, one for each vector component.

I seem to recall that's how Maxwell himself wrote them.

That agrees with the paper by Waser (http://www.aw-verlag.ch/Documents/Notation%20of%20Maxwell%20Field%20Equations.PDF). He says there were orginally 8 equations, but the first 6 have 3 components each, for a total of 18 components with the final 2 equations adding 2 more to the total.

--- Argo

[Argo Nimbus]

Originally posted by GreyWanderer
Could anyone tell me the story behind these equations? How could he come up with these complex equations that could be simplified into two?

I'll have to let someone else explain how Maxwell came up with these equations, but the simplification seems to be mostly a matter of nomenclature. Waser says of the original 8 equations that only three are "Maxwell" equations:


On the Notation of Maxwell's Field Equations, by Andre Waser (http://www.aw-verlag.ch/Documents/Notation%20of%20Maxwell%20Field%20Equations.PDF)
Three Maxwell equations can be found quickly in the original set, together with Ohm's law (1.6), the Faraday-force (1.4), and the continuity equation (1.8) for a region containing charges.


Of the "three" Maxwell equations, the one for Vector Potential is often eliminated.


The original equations do not strictly correspond to today's vector equations. The original equations, for example, contains the vector potential A, which today usually is eliminated.


This would leave "two" Maxwell equations from the original set.

--- Argo

[BeholdTheTruth]

Originally posted by Argo Nimbus


Maxwell originally worked with quaternions, so searching for maxwell + quaternions will produce some interesting links. In any case, Maxwell's original equations can be found in a pdf document here:

On the Notation of Maxwell's Field Equations, by Andre Waser (http://www.aw-verlag.ch/Documents/Notation%20of%20Maxwell%20Field%20Equations.PDF)

--- Argo


Thanks, Argo. Very helpful. Regards, Yale

[BeholdTheTruth]

Originally posted by peptoabysmal
Why do you want Maxwell's original equations, Yalel? Looking perhaps for some "divine inspiration" to mathematically prove the existance of G.O.D.?

I see you got some new suckers to take the bait.

Let he who hath understanding, understand that this thread will lead nowhere except to some pointers to a very poorly made website with gibberish content.

Pepto, let he who hath no understanding understand that Argo (by providing useful info) hath the temperament of a Maxwell or a Heaviside or a Boltzmann. You at best hath the temperament of a Mach, and at worst that of a Fool.

[MRC_Hans]

While not diplomatically worded, the question is still quite relevant, Yalel. You do have a bit of a history here, and your quickness to pronounce a critic a fool does little to change that.

I have wondered myself where this was heading. The Maxwell Equations are rather heavy math, and based on previous experience, I would guess that you are slightly out of bounds here, so if you ask me (which you probably won't ;) ), I would discourage you from attempting to make any creative conclusions from them, as these are likely to be pure speculation.

Hans

[BeholdTheTruth]

You're right. I am not asking you.

And I stand on what I said: Argo was VERY helpful, whereas Pepto (and you) not.

Case closed -- unless you can't let go.

[Crossbow]

This is not the first time Yalel has attempted to discuss Maxwell; on September 18, 2002 Yalel wrote when he was discussing the "Per Quarter Turns" topic.

...

Which is why, guys, I originally began with the Religion and Philosophy section. Notice this interesting additonal comment by the reviewer: "Lindley's account of Maxwell's religion and his statement that Maxwell "kept science and religion separate" struck me as dubious...” As dubious as Cantor, Newton, Kepler... at least to someone like me who believes in prototheism.

...

[BeholdTheTruth]

Originally posted by Crossbow
This is not the first time Yalel has attempted to discuss Maxwell; on September 18, 2002 Yalel wrote when he was discussing the "Per Quarter Turns" topic.

...

Which is why, guys, I originally began with the Religion and Philosophy section. Notice this interesting additonal comment by the reviewer: "Lindley's account of Maxwell's religion and his statement that Maxwell "kept science and religion separate" struck me as dubious...” As dubious as Cantor, Newton, Kepler... at least to someone like me who believes in prototheism.

...


CB, I am surprised that after so long a time my seemingly impotent memes still have so much power over you. E.g., many months later they seem to have engendered within you the need to quote me about something that has nothing to do with the simple and direct science question asked by me and answered just as simply and directly by Argo.

BTW, although you as usual quote me out of context, here for those interested in the histories of scientists is a summary of the Lindley book...

Boltzmann's Atom: The Great Debate that Launched a Revolution in Physics
David Lindley
Free Press, New York, 2001. $24.00 (260 pp.). ISBN 0-684-85186-5
Reviewed by Clayton A. Gearhart
The Austrian theoretical physicist Ludwig Boltzmann shaped much of the physics of the 20th century. His influence was central to Max Planck's 1900-1901 papers on blackbody radiation, to Josiah Willard Gibbs's 1902 formulation of statistical mechanics, and to Albert Einstein's 1905 papers on the light quantum and on Brownian motion. Boltzmann was not only one of the most creative and influential physicists of the 19th century, but one of its most colorful personalities as well. His amusing account of his visit to California in 1905, "Journey of a German Professor to Eldorado," has been widely reprinted, and the number and variety of Boltzmann stories rival those told about Richard Feynman!

David Lindley's new popular biography is thus a welcome addition to the Boltzmann literature. Lindley is a physicist with wide experience as an editor and science writer. He gives a solid account of Boltzmann's life, one that makes good use of the collections of Boltzmann's correspondence that have appeared in recent years. There are also thumbnail sketches of such figures as Ernst Mach, James Clerk Maxwell, and Gibbs.

It is no easy matter to present Boltzmann's scientific achievements to general readers, and here the record is mixed: Lindley gives a good account of the 1877 paper that related entropy to probability--the principle that Planck later wrote down in the famous equation S = k log W. The section on the H-theorem is perhaps a little less successful (hardly a criticism in view of its difficulty). There is also a fine account of Boltzmann's visit to England for the 1894 British Association meeting, where he found the reception of the H-theorem and kinetic theory generally to be considerably more favorable than elsewhere in Europe. There is, however, only a brief mention of Boltzmann's papers of 1871 and later, in which he developed the formalism of equilibrium statistical mechanics that Gibbs later put to such good use.

Lindley does a good job of describing the uneven reception of kinetic theory in Germany. The criticisms of Josef Loschmidt and Ernst Zermelo, based respectively on the time-reversibility of Newton's laws and on Poincaré's recurrence theorem, forced Boltzmann to clarify the probabilistic nature of both the H-theorem and the second law of thermodyamics. Other objections from the energeticists and from Mach were less fruitful. Boltzmann found all of them discouraging.

These criticisms, along with failing eyesight and other health problems (including increasingly serious if intermittent depression), plagued his later years. Lindley shows how the controversy over kinetic theory, and more generally over the role and scope of theoretical physics, eventually led Boltzmann to write and lecture on the philosophy of science. Here again, however, the author's discussion is a little too brief to give a comprehensive picture. Readers looking for a more detailed and nuanced account of Boltzmann's scientific and philosophical writings may wish to consult the recent works of Carlo Cercignani (Ludwig Boltzmann: The Man Who Trusted Atoms, Oxford U. Press, 1998) and John Blackmore, editor, (Ludwig Boltzmann, His Later Life and Philosophy, 1900-1906, Kluwer, 1995).

There are some questionable statements and interpretations that readers should be aware of. For example, there were no "later editions" of Boltzmann's 1896-1898 Lectures on Gas Theory. The Franco-Prussian War of 1870-71 was considerably more than a "skirmish." Lindley's account of Maxwell's religion and his statement that Maxwell "kept science and religion separate" struck me as dubious, as did his suggestion that Gibbs was in some sense neutral in the late 19th-century controversy over the existence of atoms and molecules. One could cite other examples. None is particularly serious--general readers will not be badly misled, and others will readily encounter corrections--but they are in sum unfortunate.

These problems should in no way discourage readers from a well-written and informative popular biography that fills a real gap in the Boltzmann literature. It serves as a fine account for general readers and a good starting point for those who want to go more deeply into Boltzmann's life and thought.



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Clayton A. Gearhart is a professor of physics at St. John's University in Collegeville, Minnesota. His research interests include the history of thermodynamics and statistical mechanics.